Welcome to Android Central! What phone are you using? Are you using the stock Messaging app, or the Hangouts app? What happens when you tap on the photo--does it expand to fill the screen? Then tap it again and press the Menu button (the 3 vertical dot icon that is probably in the upper right corner), and press Save.
Alternatively, you might have to long-press the photo (i.e., press and hold until an option comes up).